Rosa's Benchmark

Percentage Accurate: 99.8% → 99.8%

Time: 8.0s

Alternatives: 7

Speedup: 1.2×

• 34.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))
end function

Java

public static double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Python

def code(x):
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))

C

double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (0.954929658551372d0 * x) - (0.12900613773279798d0 * ((x * x) * x))
end function

Java

public static double code(double x) {
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
}

Python

def code(x):
	return (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x))

Julia

function code(x)
	return Float64(Float64(0.954929658551372 * x) - Float64(0.12900613773279798 * Float64(Float64(x * x) * x)))
end

MATLAB

function tmp = code(x)
	tmp = (0.954929658551372 * x) - (0.12900613773279798 * ((x * x) * x));
end

Wolfram

code[x_] := N[(N[(0.954929658551372 * x), $MachinePrecision] - N[(0.12900613773279798 * N[(N[(x * x), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x \cdot x, x \cdot -0.12900613773279798, x \cdot 0.954929658551372\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (* x x) (* x -0.12900613773279798) (* x 0.954929658551372)))

C

double code(double x) {
	return fma((x * x), (x * -0.12900613773279798), (x * 0.954929658551372));
}

Julia

function code(x)
	return fma(Float64(x * x), Float64(x * -0.12900613773279798), Float64(x * 0.954929658551372))
end

Wolfram

code[x_] := N[(N[(x * x), $MachinePrecision] * N[(x * -0.12900613773279798), $MachinePrecision] + N[(x * 0.954929658551372), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Step-by-step derivation

   1. cancel-sign-sub-inv 99.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x + \left(-0.12900613773279798\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

   2. metadata-eval 99.8%

      \[\leadsto 0.954929658551372 \cdot x + \color{blue}{-0.12900613773279798} \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   3. pow3 99.7%

      \[\leadsto 0.954929658551372 \cdot x + -0.12900613773279798 \cdot \color{blue}{{x}^{3}} \]

   4. +-commutative 99.7%

      \[\leadsto \color{blue}{-0.12900613773279798 \cdot {x}^{3} + 0.954929658551372 \cdot x} \]

   5. pow3 99.8%

      \[\leadsto -0.12900613773279798 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + 0.954929658551372 \cdot x \]

   6. *-commutative 99.8%

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot -0.12900613773279798} + 0.954929658551372 \cdot x \]

   7. associate-*l* 99.8%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)} + 0.954929658551372 \cdot x \]

   8. fma-def 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot -0.12900613773279798, 0.954929658551372 \cdot x\right)} \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot -0.12900613773279798, 0.954929658551372 \cdot x\right)} \]

4. Final simplification 99.8%

   \[\leadsto \mathsf{fma}\left(x \cdot x, x \cdot -0.12900613773279798, x \cdot 0.954929658551372\right) \]

Alternative 2: 98.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (or (<= x -2.7) (not (<= x 2.7)))
   (* x (* (* x x) -0.12900613773279798))
   (* x 0.954929658551372)))

C

double code(double x) {
	double tmp;
	if ((x <= -2.7) || !(x <= 2.7)) {
		tmp = x * ((x * x) * -0.12900613773279798);
	} else {
		tmp = x * 0.954929658551372;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((x <= (-2.7d0)) .or. (.not. (x <= 2.7d0))) then
        tmp = x * ((x * x) * (-0.12900613773279798d0))
    else
        tmp = x * 0.954929658551372d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((x <= -2.7) || !(x <= 2.7)) {
		tmp = x * ((x * x) * -0.12900613773279798);
	} else {
		tmp = x * 0.954929658551372;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (x <= -2.7) or not (x <= 2.7):
		tmp = x * ((x * x) * -0.12900613773279798)
	else:
		tmp = x * 0.954929658551372
	return tmp

Julia

function code(x)
	tmp = 0.0
	if ((x <= -2.7) || !(x <= 2.7))
		tmp = Float64(x * Float64(Float64(x * x) * -0.12900613773279798));
	else
		tmp = Float64(x * 0.954929658551372);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((x <= -2.7) || ~((x <= 2.7)))
		tmp = x * ((x * x) * -0.12900613773279798);
	else
		tmp = x * 0.954929658551372;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Or[LessEqual[x, -2.7], N[Not[LessEqual[x, 2.7]], $MachinePrecision]], N[(x * N[(N[(x * x), $MachinePrecision] * -0.12900613773279798), $MachinePrecision]), $MachinePrecision], N[(x * 0.954929658551372), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -2.7000000000000002 or 2.7000000000000002 < x

   1. Initial program 99.8%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
   2. Step-by-step derivation

      1. sqr-neg 99.8%

         \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)} \cdot x\right) \]

      2. associate-*r* 99.8%

         \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(\left(-x\right) \cdot \left(-x\right)\right)\right) \cdot x} \]

      3. distribute-rgt-out-- 99.8%

         \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(\left(-x\right) \cdot \left(-x\right)\right)\right)} \]

      4. sub-neg 99.8%

         \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798 \cdot \left(\left(-x\right) \cdot \left(-x\right)\right)\right)\right)} \]

      5. +-commutative 99.8%

         \[\leadsto x \cdot \color{blue}{\left(\left(-0.12900613773279798 \cdot \left(\left(-x\right) \cdot \left(-x\right)\right)\right) + 0.954929658551372\right)} \]

      6. associate-*r* 99.8%

         \[\leadsto x \cdot \left(\left(-\color{blue}{\left(0.12900613773279798 \cdot \left(-x\right)\right) \cdot \left(-x\right)}\right) + 0.954929658551372\right) \]

      7. distribute-rgt-neg-in 99.8%

         \[\leadsto x \cdot \left(\color{blue}{\left(0.12900613773279798 \cdot \left(-x\right)\right) \cdot \left(-\left(-x\right)\right)} + 0.954929658551372\right) \]

      8. remove-double-neg 99.8%

         \[\leadsto x \cdot \left(\left(0.12900613773279798 \cdot \left(-x\right)\right) \cdot \color{blue}{x} + 0.954929658551372\right) \]

      9. *-commutative 99.8%

         \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(0.12900613773279798 \cdot \left(-x\right)\right)} + 0.954929658551372\right) \]

      10. fma-def 99.8%

          \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 0.12900613773279798 \cdot \left(-x\right), 0.954929658551372\right)} \]

      11. distribute-rgt-neg-out 99.8%

          \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{-0.12900613773279798 \cdot x}, 0.954929658551372\right) \]

      12. distribute-lft-neg-in 99.8%

          \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(-0.12900613773279798\right) \cdot x}, 0.954929658551372\right) \]

      13. *-commutative 99.8%

          \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12900613773279798\right)}, 0.954929658551372\right) \]

      14. metadata-eval 99.8%

          \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot -0.12900613773279798, 0.954929658551372\right)} \]
   4. Step-by-step derivation

      1. fma-udef 99.8%

         \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot -0.12900613773279798\right) + 0.954929658551372\right)} \]

   5. Applied egg-rr 99.8%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot -0.12900613773279798\right) + 0.954929658551372\right)} \]
   6. Step-by-step derivation

      1. +-commutative 99.8%

         \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + x \cdot \left(x \cdot -0.12900613773279798\right)\right)} \]

      2. flip-+ 36.4%

         \[\leadsto x \cdot \color{blue}{\frac{0.954929658551372 \cdot 0.954929658551372 - \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right) \cdot \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)}{0.954929658551372 - x \cdot \left(x \cdot -0.12900613773279798\right)}} \]

      3. metadata-eval 36.4%

         \[\leadsto x \cdot \frac{\color{blue}{0.9118906527810399} - \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right) \cdot \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)}{0.954929658551372 - x \cdot \left(x \cdot -0.12900613773279798\right)} \]

      4. associate-*r* 36.4%

         \[\leadsto x \cdot \frac{0.9118906527810399 - \color{blue}{\left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)} \cdot \left(x \cdot \left(x \cdot -0.12900613773279798\right)\right)}{0.954929658551372 - x \cdot \left(x \cdot -0.12900613773279798\right)} \]

      5. associate-*r* 36.3%

         \[\leadsto x \cdot \frac{0.9118906527810399 - \left(\left(x \cdot x\right) \cdot -0.12900613773279798\right) \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)}}{0.954929658551372 - x \cdot \left(x \cdot -0.12900613773279798\right)} \]

      6. swap-sqr 35.4%

         \[\leadsto x \cdot \frac{0.9118906527810399 - \color{blue}{\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \left(-0.12900613773279798 \cdot -0.12900613773279798\right)}}{0.954929658551372 - x \cdot \left(x \cdot -0.12900613773279798\right)} \]

      7. metadata-eval 35.4%

         \[\leadsto x \cdot \frac{0.9118906527810399 - \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \color{blue}{0.016642583572733644}}{0.954929658551372 - x \cdot \left(x \cdot -0.12900613773279798\right)} \]

      8. pow2 35.4%

         \[\leadsto x \cdot \frac{0.9118906527810399 - \left(\color{blue}{{x}^{2}} \cdot \left(x \cdot x\right)\right) \cdot 0.016642583572733644}{0.954929658551372 - x \cdot \left(x \cdot -0.12900613773279798\right)} \]

      9. pow2 35.4%

         \[\leadsto x \cdot \frac{0.9118906527810399 - \left({x}^{2} \cdot \color{blue}{{x}^{2}}\right) \cdot 0.016642583572733644}{0.954929658551372 - x \cdot \left(x \cdot -0.12900613773279798\right)} \]

      10. pow-prod-up 35.5%

          \[\leadsto x \cdot \frac{0.9118906527810399 - \color{blue}{{x}^{\left(2 + 2\right)}} \cdot 0.016642583572733644}{0.954929658551372 - x \cdot \left(x \cdot -0.12900613773279798\right)} \]

      11. metadata-eval 35.5%

          \[\leadsto x \cdot \frac{0.9118906527810399 - {x}^{\color{blue}{4}} \cdot 0.016642583572733644}{0.954929658551372 - x \cdot \left(x \cdot -0.12900613773279798\right)} \]

      12. associate-*r* 35.6%

          \[\leadsto x \cdot \frac{0.9118906527810399 - {x}^{4} \cdot 0.016642583572733644}{0.954929658551372 - \color{blue}{\left(x \cdot x\right) \cdot -0.12900613773279798}} \]

   7. Applied egg-rr 35.6%

      \[\leadsto x \cdot \color{blue}{\frac{0.9118906527810399 - {x}^{4} \cdot 0.016642583572733644}{0.954929658551372 - \left(x \cdot x\right) \cdot -0.12900613773279798}} \]

   8. Taylor expanded in x around inf 98.8%

      \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot {x}^{2}\right)} \]
   9. Step-by-step derivation

      1. unpow2 98.8%

         \[\leadsto x \cdot \left(-0.12900613773279798 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   10. Simplified 98.8%

       \[\leadsto x \cdot \color{blue}{\left(-0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

   if -2.7000000000000002 < x < 2.7000000000000002

   1. Initial program 99.7%

      \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   2. Taylor expanded in x around 0 99.4%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
   3. Step-by-step derivation

      1. *-commutative 99.4%

         \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]

   4. Simplified 99.4%

      \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \lor \neg \left(x \leq 2.7\right):\\ \;\;\;\;x \cdot \left(\left(x \cdot x\right) \cdot -0.12900613773279798\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot 0.954929658551372\\ \end{array} \]

Alternative 3: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right) + x \cdot 0.954929658551372 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (+ (* (* x x) (* x -0.12900613773279798)) (* x 0.954929658551372)))

C

double code(double x) {
	return ((x * x) * (x * -0.12900613773279798)) + (x * 0.954929658551372);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = ((x * x) * (x * (-0.12900613773279798d0))) + (x * 0.954929658551372d0)
end function

Java

public static double code(double x) {
	return ((x * x) * (x * -0.12900613773279798)) + (x * 0.954929658551372);
}

Python

def code(x):
	return ((x * x) * (x * -0.12900613773279798)) + (x * 0.954929658551372)

Julia

function code(x)
	return Float64(Float64(Float64(x * x) * Float64(x * -0.12900613773279798)) + Float64(x * 0.954929658551372))
end

MATLAB

function tmp = code(x)
	tmp = ((x * x) * (x * -0.12900613773279798)) + (x * 0.954929658551372);
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * N[(x * -0.12900613773279798), $MachinePrecision]), $MachinePrecision] + N[(x * 0.954929658551372), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Step-by-step derivation

   1. cancel-sign-sub-inv 99.8%

      \[\leadsto \color{blue}{0.954929658551372 \cdot x + \left(-0.12900613773279798\right) \cdot \left(\left(x \cdot x\right) \cdot x\right)} \]

   2. metadata-eval 99.8%

      \[\leadsto 0.954929658551372 \cdot x + \color{blue}{-0.12900613773279798} \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

   3. pow3 99.7%

      \[\leadsto 0.954929658551372 \cdot x + -0.12900613773279798 \cdot \color{blue}{{x}^{3}} \]

   4. +-commutative 99.7%

      \[\leadsto \color{blue}{-0.12900613773279798 \cdot {x}^{3} + 0.954929658551372 \cdot x} \]

   5. pow3 99.8%

      \[\leadsto -0.12900613773279798 \cdot \color{blue}{\left(\left(x \cdot x\right) \cdot x\right)} + 0.954929658551372 \cdot x \]

   6. *-commutative 99.8%

      \[\leadsto \color{blue}{\left(\left(x \cdot x\right) \cdot x\right) \cdot -0.12900613773279798} + 0.954929658551372 \cdot x \]

   7. associate-*l* 99.8%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right)} + 0.954929658551372 \cdot x \]

   8. fma-def 99.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot -0.12900613773279798, 0.954929658551372 \cdot x\right)} \]

3. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x \cdot x, x \cdot -0.12900613773279798, 0.954929658551372 \cdot x\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.8%

      \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right) + 0.954929658551372 \cdot x} \]

   2. *-commutative 99.8%

      \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right) + \color{blue}{x \cdot 0.954929658551372} \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right) + x \cdot 0.954929658551372} \]

6. Final simplification 99.8%

   \[\leadsto \left(x \cdot x\right) \cdot \left(x \cdot -0.12900613773279798\right) + x \cdot 0.954929658551372 \]

Alternative 4: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.954929658551372 + x \cdot \left(x \cdot -0.12900613773279798\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 0.954929658551372 (* x (* x -0.12900613773279798)))))

C

double code(double x) {
	return x * (0.954929658551372 + (x * (x * -0.12900613773279798)));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.954929658551372d0 + (x * (x * (-0.12900613773279798d0))))
end function

Java

public static double code(double x) {
	return x * (0.954929658551372 + (x * (x * -0.12900613773279798)));
}

Python

def code(x):
	return x * (0.954929658551372 + (x * (x * -0.12900613773279798)))

Julia

function code(x)
	return Float64(x * Float64(0.954929658551372 + Float64(x * Float64(x * -0.12900613773279798))))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.954929658551372 + (x * (x * -0.12900613773279798)));
end

Wolfram

code[x_] := N[(x * N[(0.954929658551372 + N[(x * N[(x * -0.12900613773279798), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Step-by-step derivation

   1. sqr-neg 99.8%

      \[\leadsto 0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\color{blue}{\left(\left(-x\right) \cdot \left(-x\right)\right)} \cdot x\right) \]

   2. associate-*r* 99.7%

      \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(\left(-x\right) \cdot \left(-x\right)\right)\right) \cdot x} \]

   3. distribute-rgt-out-- 99.7%

      \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(\left(-x\right) \cdot \left(-x\right)\right)\right)} \]

   4. sub-neg 99.7%

      \[\leadsto x \cdot \color{blue}{\left(0.954929658551372 + \left(-0.12900613773279798 \cdot \left(\left(-x\right) \cdot \left(-x\right)\right)\right)\right)} \]

   5. +-commutative 99.7%

      \[\leadsto x \cdot \color{blue}{\left(\left(-0.12900613773279798 \cdot \left(\left(-x\right) \cdot \left(-x\right)\right)\right) + 0.954929658551372\right)} \]

   6. associate-*r* 99.7%

      \[\leadsto x \cdot \left(\left(-\color{blue}{\left(0.12900613773279798 \cdot \left(-x\right)\right) \cdot \left(-x\right)}\right) + 0.954929658551372\right) \]

   7. distribute-rgt-neg-in 99.7%

      \[\leadsto x \cdot \left(\color{blue}{\left(0.12900613773279798 \cdot \left(-x\right)\right) \cdot \left(-\left(-x\right)\right)} + 0.954929658551372\right) \]

   8. remove-double-neg 99.7%

      \[\leadsto x \cdot \left(\left(0.12900613773279798 \cdot \left(-x\right)\right) \cdot \color{blue}{x} + 0.954929658551372\right) \]

   9. *-commutative 99.7%

      \[\leadsto x \cdot \left(\color{blue}{x \cdot \left(0.12900613773279798 \cdot \left(-x\right)\right)} + 0.954929658551372\right) \]

   10. fma-def 99.7%

       \[\leadsto x \cdot \color{blue}{\mathsf{fma}\left(x, 0.12900613773279798 \cdot \left(-x\right), 0.954929658551372\right)} \]

   11. distribute-rgt-neg-out 99.7%

       \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{-0.12900613773279798 \cdot x}, 0.954929658551372\right) \]

   12. distribute-lft-neg-in 99.7%

       \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{\left(-0.12900613773279798\right) \cdot x}, 0.954929658551372\right) \]

   13. *-commutative 99.7%

       \[\leadsto x \cdot \mathsf{fma}\left(x, \color{blue}{x \cdot \left(-0.12900613773279798\right)}, 0.954929658551372\right) \]

   14. metadata-eval 99.7%

       \[\leadsto x \cdot \mathsf{fma}\left(x, x \cdot \color{blue}{-0.12900613773279798}, 0.954929658551372\right) \]

3. Simplified 99.7%

   \[\leadsto \color{blue}{x \cdot \mathsf{fma}\left(x, x \cdot -0.12900613773279798, 0.954929658551372\right)} \]
4. Step-by-step derivation

   1. fma-udef 99.7%

      \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot -0.12900613773279798\right) + 0.954929658551372\right)} \]

5. Applied egg-rr 99.7%

   \[\leadsto x \cdot \color{blue}{\left(x \cdot \left(x \cdot -0.12900613773279798\right) + 0.954929658551372\right)} \]

6. Final simplification 99.7%

   \[\leadsto x \cdot \left(0.954929658551372 + x \cdot \left(x \cdot -0.12900613773279798\right)\right) \]

Alternative 5: 99.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(0.954929658551372 - \left(x \cdot x\right) \cdot 0.12900613773279798\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (- 0.954929658551372 (* (* x x) 0.12900613773279798))))

C

double code(double x) {
	return x * (0.954929658551372 - ((x * x) * 0.12900613773279798));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (0.954929658551372d0 - ((x * x) * 0.12900613773279798d0))
end function

Java

public static double code(double x) {
	return x * (0.954929658551372 - ((x * x) * 0.12900613773279798));
}

Python

def code(x):
	return x * (0.954929658551372 - ((x * x) * 0.12900613773279798))

Julia

function code(x)
	return Float64(x * Float64(0.954929658551372 - Float64(Float64(x * x) * 0.12900613773279798)))
end

MATLAB

function tmp = code(x)
	tmp = x * (0.954929658551372 - ((x * x) * 0.12900613773279798));
end

Wolfram

code[x_] := N[(x * N[(0.954929658551372 - N[(N[(x * x), $MachinePrecision] * 0.12900613773279798), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]
2. Step-by-step derivation

   1. associate-*r* 99.7%

      \[\leadsto 0.954929658551372 \cdot x - \color{blue}{\left(0.12900613773279798 \cdot \left(x \cdot x\right)\right) \cdot x} \]

   2. distribute-rgt-out-- 99.7%

      \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

3. Applied egg-rr 99.7%

   \[\leadsto \color{blue}{x \cdot \left(0.954929658551372 - 0.12900613773279798 \cdot \left(x \cdot x\right)\right)} \]

4. Final simplification 99.7%

   \[\leadsto x \cdot \left(0.954929658551372 - \left(x \cdot x\right) \cdot 0.12900613773279798\right) \]

Alternative 6: 5.1% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ x \cdot -0.954929658551372 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x -0.954929658551372))

C

double code(double x) {
	return x * -0.954929658551372;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (-0.954929658551372d0)
end function

Java

public static double code(double x) {
	return x * -0.954929658551372;
}

Python

def code(x):
	return x * -0.954929658551372

Julia

function code(x)
	return Float64(x * -0.954929658551372)
end

MATLAB

function tmp = code(x)
	tmp = x * -0.954929658551372;
end

Wolfram

code[x_] := N[(x * -0.954929658551372), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

2. Taylor expanded in x around 0 52.6%

   \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
3. Step-by-step derivation

   1. *-commutative 52.6%

      \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]

4. Simplified 52.6%

   \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]
5. Step-by-step derivation

   1. add-sqr-sqrt 24.7%

      \[\leadsto \color{blue}{\left(\sqrt{x} \cdot \sqrt{x}\right)} \cdot 0.954929658551372 \]

   2. sqrt-prod 27.1%

      \[\leadsto \color{blue}{\sqrt{x \cdot x}} \cdot 0.954929658551372 \]

   3. metadata-eval 27.1%

      \[\leadsto \sqrt{x \cdot x} \cdot \color{blue}{\sqrt{0.9118906527810399}} \]

   4. sqrt-prod 27.1%

      \[\leadsto \color{blue}{\sqrt{\left(x \cdot x\right) \cdot 0.9118906527810399}} \]

   5. associate-*l* 27.1%

      \[\leadsto \sqrt{\color{blue}{x \cdot \left(x \cdot 0.9118906527810399\right)}} \]

6. Applied egg-rr 27.1%

   \[\leadsto \color{blue}{\sqrt{x \cdot \left(x \cdot 0.9118906527810399\right)}} \]

7. Taylor expanded in x around -inf 5.0%

   \[\leadsto \color{blue}{-0.954929658551372 \cdot x} \]
8. Step-by-step derivation

   1. *-commutative 5.0%

      \[\leadsto \color{blue}{x \cdot -0.954929658551372} \]

9. Simplified 5.0%

   \[\leadsto \color{blue}{x \cdot -0.954929658551372} \]

10. Final simplification 5.0%

    \[\leadsto x \cdot -0.954929658551372 \]

Alternative 7: 49.4% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ x \cdot 0.954929658551372 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x 0.954929658551372))

C

double code(double x) {
	return x * 0.954929658551372;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * 0.954929658551372d0
end function

Java

public static double code(double x) {
	return x * 0.954929658551372;
}

Python

def code(x):
	return x * 0.954929658551372

Julia

function code(x)
	return Float64(x * 0.954929658551372)
end

MATLAB

function tmp = code(x)
	tmp = x * 0.954929658551372;
end

Wolfram

code[x_] := N[(x * 0.954929658551372), $MachinePrecision]

Derivation

1. Initial program 99.8%

   \[0.954929658551372 \cdot x - 0.12900613773279798 \cdot \left(\left(x \cdot x\right) \cdot x\right) \]

2. Taylor expanded in x around 0 52.6%

   \[\leadsto \color{blue}{0.954929658551372 \cdot x} \]
3. Step-by-step derivation

   1. *-commutative 52.6%

      \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]

4. Simplified 52.6%

   \[\leadsto \color{blue}{x \cdot 0.954929658551372} \]

5. Final simplification 52.6%

   \[\leadsto x \cdot 0.954929658551372 \]

Reproduce

herbie shell --seed 2023271 
(FPCore (x)
  :name "Rosa's Benchmark"
  :precision binary64
  (- (* 0.954929658551372 x) (* 0.12900613773279798 (* (* x x) x))))